'« 



NON-ARISTOTELIAN LOGIC 



BY 



HENRY BRADFORD SMITH 

Assistant Professor of Philosophy in the University of Pennsylvania 



THE COLLEGE BOOK STORE 

3425 WOODLAND AVENUE 

PHILADELPHIA, PA. 

1919 






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5 



PRESS OF 

THE NEW ERA PRINTING COMPANY 

LANCASTER, PA. 



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PREFACE 



In the writer's Primer of Logic (B. D. Smith and Bros., 
Pulaski, Va., 191 7) the classical science is treated as if it 
were a special case of a more general logic, from which it is 
distinguished by certain postulates. This work follows an 
established tradition, which regards the proposition, No a 
is non-a, as true for all meanings of a. But the Aristotelian 
system is not, following the same tradition, regarded as 
false, but is viewed as a logic of limited application, whose 
terms can not take on the values nothing and universe. 

An article of later date, N on- Aristotelian Logic (Journal 
of Philosophy, Psychology and Scientific Methods, August 
15, 19 1 8), has shown the possibility of an extension of the 
classical logic, which would allow all of its implications 
to remain true without any restriction being imposed upon 
the meanings of the terms. Furthermore, the existence of 
a new family of logics is there indicated and it is pointed 
out that each member of this group finds its application in 
precisely the same world of common experience. 

A paper, On the Construction of a Non-Aristotelian Logic 
(Monist, July, 191 8) has set forth in detail one of these 
systems, whose characteristic postulate is most at variance 
with ordinary intuition. These results and others are here 
brought together in more complete and systematic form. 

I have again to express my indebtedness to Professor 
E. A. Singer, Jr., acknowledged in the preface to the first 
work mentioned above, for my introduction to the method, 
which is here employed. My Letters on Logic to a Young 
Man without a Master y which is about to appear, will contain 
a popular introduction to Professor Singer's system. 

H. B. S. 



m 



TABLE OF CONTENTS 
CHAPTER I 

PAGES 

§§ 1-14. Foundations of the common logic. 
The forms of immediate inference. 
Exercises I_I 3 

CHAPTER II 

§§ 15-20. Deduction of the valid and the invalid 
moods of the syllogism. 
Exercises 14-21 

CHAPTER III 

§§ 21-23. Construction of a Non-Aristotelian logic. 

Exercises 22-25 

CHAPTER IV 

§§ 24-25. The general solution of the sorites. 

Exercises 26-36 

CHAPTER V 

§ 26. Alternative systems. 

Exercises 37~40 



NON-ARISTOTELIAN LOGIC 



CHAPTER I 

§ i. The problem of a science is to define the elements, 
of which it treats, by an enumeration of their formal 
properties. These properties are to be found within the 
system, of which these elements are the parts. The task 
of logic is, then, to develop completely its own system, by 
constructing all the true and all the untrue propositions, 
into which its elements enter exclusively. 

§ 2. The propositions, which are recognized by the logi- 
cian, are: 

(i) The Categorical forms, made up of terms (represented 
in the proposition x(ab) by the subject a and the predicate 
b) and relationships (an adjective of quantity, all, some, 
no, and the copula, is), viz., 

a(ab) = All a is all b, 

(3(ab) = Some a is some b, 

y(ab) — All a is some b, 

e(ab) = No a is b, 

the word some, explicit in (3 and y, being interpreted to 
mean some at least, not all. This meaning of the word will 
be unambiguously determined by the propositions, which 
we say shall be true or untrue in our science. Whenever 
we wish to designate some one of the four forms but desire 
to leave unsettled which one is meant, we shall employ the 
notation, x(ab), y(ab), z(ab), etc. 
(2) The Hypothetical forms, 

x(ab) z y(ab) = x(ab) implies y(ab) is true, 
{x{ab) z y(a>b)Y = x(ab) implies y(ab) is untrue. 



2 Non-Aristotelian Logic 

(3) The Conjunctive form, 

x(ab)-y(ab) = x(ab) and y(ab) are true, 

(4) The Disjunctive form, 

x(ab) + y(ab) = x(ab) or y{ab) is true. 

x(ab) is untrue will be represented by x'iab), x f {ab) is 
untrue by x ,r {ab), etc. Since in the proposition x(ab) the 
terms are subject, a, and predicate, b, the term-order is the 
order subject-predicate. Whenever we wish to leave the 
term-order unsettled we shall place a comma between the 
terms. Thus, x{a, b) stands for x(ab) or for x{ba). 

§ 3. A principle, which is altogether fundamental and 
which will be taken for granted at each step of our progress, 
is this: If a proposition is true in general, it is because it 
remains true for all specific meanings of the terms that enter 
into it, although an untrue proposition may well enough be- 
come true in the same circumstances. Thus y{ab) z y (ba) 
is untrue in general, but it becomes true for the case, in 
which a and b are identical, viz., y(aa) z y(aa). Accord- 
ingly, when we write {x(a, b) z y(a, b)} f , we only assert 
that there is at least one value of a and one value of b, 
which will render x(a, b) z y(a, b) a false proposition. If 
it has been established that x{aa) z y(aa) is untrue, then 
we may at once infer that the more general implication, 
x(a, b) z y(a, b), is untrue as well. In order to establish the 
untruth of a given proposition, it will be enough to point to a 
special instance of its being untrue. 

§ 4. In presenting the materials of our subject-matter we 
shall have to deal with two types of proposition. The 
truth of 

(x z y)(y z z) z (x z z) 

is independent of x, y and z no matter for what propositions 
x, y and z may stand. Such a general truth will be termed 
a principle. The truth of 

y(ba)y(cb) z y(ca) 



Foundations of the Common Logic 3 

is independent of a, b and c no matter for what classes a, b 
and c may stand. If such a general truth has to be taken 
for granted, it will be termed a postulate. 

Principles are, accordingly, independent of forms] postu- 
lates are independent of terms. 

§ 5. We begin by setting down four postulates, the truth 
of which may be verified at once empirically by the familiar 
device of Euler's circular diagrams, 

(i) a(ba)a(bc) Z a(ca), 

(ii) a{ba)p{bc) z 0(ca), 

(iii) a(ba)y{cb) z y(ca), 

(iv) a(ba) e(bc) Z e(ca), 

and we shall add to these, 

(v) a r (aa) / a(aa). 

This last assumption illustrates an extension of the common 
meaning of implication and is forced upon us, if we are to 
allow a(aa) to stand for a true proposition. The uses, to 
which such an extension of meaning may be put, will be- 
come clear in the sequel. It will be enough to state that 
a proposition, which is true for all meanings of the terms, 
will be implied by the proposition whose symbol is i, and 
behaves like a unit multiplier in this algebra. 

Only a small number of the principles, which we shall 
introduce as necessity requires, are independent but it will 
not concern our purpose to point out the manner of their 
inter-connection . 

From the principle, 

(*' Z x) Z (y Z x), 
we obtain, by (v), the theorem, 

i z a(aa). 

By (i), for a = b, a(aa)a(ac) z a(ca), and 

{a(aa)a(ac) z a{ca)}{i z a(aa) } z {a(ac) z a(ea)}, 
by (xy z z)(w z x) z (wy z 2), omitting the unit multi- 
plier, i, as it is the custom to do. 



4 Non-Aristotelian Logic 

Similarly, we obtain 

P(ac) z (3(ca), from (ii), 

y(ca) z y(ca), from (iii), 

e(ac) z e(ca), from (iv). 
Again, 

{a(ab) z a(ba)}{a{ba) z a(ab)} z {oi(a&) z a(ab)}, 
by (x z y)(y Z z) Z (x z z). 
Accordingly, 

a(ab) z a(ab), 
P(ob) Z 0(a&), 
t(^) Z y(ab), 
e(ab) z e(a&). 

A complete induction of the members of this set and an 
application of the principle, 

(x Z y) Z (/ Z «0, 

yields the general result, 

k'(ab) z *'(a5), I 

k(ab) being understood as representing any one of the 
unprimed letters, a, (3, y, e. 

§ 6. Each one of the propositions so far derived, being 
true, is implied by the unit multiplier, i. The contra- 
dictory (as it is called) of i will be any proposition, which is 
untrue for all meanings of the terms that enter into it. It 
will be represented by the symbol, o, and will be defined by 

o Z h (i Z o)', 

wherein it will be seen that i stands for o\ For a verbal 
interpretation of o and i we may read : 

o = No proposition is true, 
i = One proposition is true. 

§ 7. The utility of the concept of zero, (the null- 
proposition), and of one, i (the <we-proposition) , will be 
illustrated in part by the derivations, which follow. We 
have 



Foundations of the Common Logic 5 

[a(ab) Z a(ab)\ z {a{ab)a{ab) Z 0}, 

by (x z y) Z 0/ Z °) I 

[a(a&)«'(a6) z 0}|z {a(a&) z «"(«&)}, 

by (ry z 0) Z (x /. y')- Thus we should obtain 

a(ab) z a"(a&), 

fcab) z /3"(a6), 

7« Z y"(ab), 

e(ab) Z €"(06), 

and, if we add to these the following assumptions, 

a"(ab) z a(ab), 

(3"(ab) z P(ab), 

y"{ab) z y(flb), 

e"(ab) Z e(o&), 

a general result will be obvious, viz., 

*(fl#) Z fc"(o&), k"(ab) z *(aft), ir 

wherein the same restriction is imposed upon k(ab) as 
before. 

§ 8. The principle, that the truth of any one of the four 
categorical forms implies the falsity of each one of the 
others, a generalization, which will now be established, is 
characteristic of the logic, which we are constructing. We 
shall begin by setting down the three characteristic postu- 
lates, 

(vi) (3(aa) z P'(aa}, 

(vii) y(aa) z y'(aa), 

(viii) e(aa) z e'(aa). 

Then, by the principle, 

(x z x f ) z (x z y), 

we may establish at once, 

P(aa) z 0, 

y(aa) z o, 

e(aa) z 0. 



6 Non-Aristotelian Logic 

Postulate (ii) above yields, for a = c,a(ba)p(ba) z fi(aa)\ 
\a{ba)p(ba) z p(aa)} {${aa) z 0} z {a{ba)fi(pa) Z o], 
by Cv z y){y z z) z (x z s); 

{a(ba)(3(ba) z 0} z W{ba) z P'(ba)}, 

by {xy z 0) z (x z /). 

Similarly, by (iii), a{ba) z y'(ab); 

\a(ab) zcc{ba)}{aiba) Z y'(ab)} z {a(ab) Z y'(ab)} } 

by (x Z y)(y Z z) z (x z z); and, by (iv), we obtain 
a(ab) z e(ab). 

If now we postulate, 

(ix) y(ba)y(cb) z y(ca), 

(x) e(ba)y(cb) z e(ca), 

there result as before e(ab) z y'{ab) and y(ab) z y'(ba), 

and if 

(xi) P(ab) z y'(ab), 

(xii) (3(ab) z e'(ab), 

all of the remaining implications of this form, viz., 

e(ab) z a'{ab), y{ab) z a'(ab), 
e(ab) z P'(ab), y(ab) z P'(ab), 
e(ab) z y'{ab), (S(ab) z a'(ab), 

are obtained at once from those that have just been estab- 
lished by 

z y') z (y z x f ). 

If, now, k(ab) and w{ab) can not represent the same 
categorical form, y(ab) and y(ba) being considered distinct, 
and if further k(ab) and w{ab) can stand only for the un- 
primed letters, a, 13, y, e, 

k(ab) z w'{ab). Ill 

This generalization is one of the characteristics, which 
marks an equivalence of the logic, whose foundations are 
here set down, with the classical logic of Aristotle. If we 
were to follow an accepted modern tradition, which regards 
e(aa') as a true proposition, (V = non-a), not all im- 



Foundations of the Common Logic 7 

plications of this type will hold, for y(ab) z e'(ab) and 
e(ab) z y'{ab) become e(oi) z y'{oi) and y{oi) z t'(oi), for 
a = 0, b = i (see § 13 below). In such a logic, which is, in- 
deed, an alternative, or Non-Aristotelian system, but which 
gives up the advantage gained by our symmetry, we should 
have to write {y(ab) z * r (ab)}' and {e(ab) z y'(ab)}'. 
Moreover, postulate (x), from which these results are 
derived becomes itself untrue and the same remark applies 
to e(ab)y(cb) z e(ca) and y(ab)e(c, b) z e(ca), implications, 
which will later be made to depend on postulate (x). 

§ 9. We shall now establish the untruth of certain forms 
of implication, making them ultimately depend upon the 
invalidity of i z o, whose untruth is set down as a matter 
of definition. 

Suppose a(aa) z <x'(aa) were true. 

{i z a(aa)} z W(aa) z o} } 
by 0' z y) Z (/ Z x) ; 

{i z a(aa)}{a(aa) z a'(aa)} z {i Z a'(aa)}, 
by (x z y)(y Z z) Z Z 2); 

{* Z a'(aa)}{ce'(>a) Z 0} Z {* Z 0}, 
by the same principle. 
But i z is untrue. 

.*. a(aa) z a!(aa) is untrue. 
Again, 

{a(aa) z a'(a#) }'{/3(a#) Z a'(aa)} Z {<*(##) z (3(aa)}', 

by (x z «)'Cy z z) z (x z y)'\ 

{a{aa) z $'(aa)}{a(aa) z P(aa)}' z {P'(aa) z P(aa)}', 

by (x Z y){x z z)' Z (y Z z)'. 

Accordingly, we have 

{a (aa) z a'(aa)}', 

lf3'(aa) z P(aa)}', 

W(aa) z y (aa)}', 

{ e'(aa) z e {aa)}'. 



8 Non-Aristotelian Logic 

and, since the untruth of any proposition is implied, when- 
ever we can point to a special instance of its being untrue, 
it follows that 

\*{ab) z a'(ab)}', 

iP'(ab) Z ftab)}', 

W{ab) z y(ab)}', 

{e'(ab) z e(ab)}', 

The first and third members of the set 

\a'(db) Z a(ab)}', 
\P(ab) Z Ptflb)}', 

(tW'zt'MI', 

{e(ab) Z e'(ab)}', 

will be established on making a = o,b = i. For the reduc- 
tion of the second and fourth see exercise (10) at the end 
of this chapter. 

As a result of a complete induction of the members of 
these sets and upon application of (x z y)' Z (y f Z x')' ', 
it follows that 

{k (ab) z V (ab)}', [k' (ab) z k (ab)}', IV 

{k'(ab) z k"(ab)}', {k"(ab) z k'(ab)}'. 

§ 10. If a'(ab) z P(ab) were a true implication, we should 
have: 

{y(ab) z €t(flb)\[ct(flb) Z j8(o6)} Z [yiflb) Z 0(a&)}, 
by (* z y)(y Z *) Z Z z); 

{7(06) Z /3(a6)H/3(aZ>) Z t'WI Z [y(ab) Z 7>&)h 
by the same principle. 

,\ a'(ab) Z &{ab) is untrue. 

Applying the same method of reduction there will result : 

{a'(ab) Z P(flb)}', \P\ab) Z y(ab)}', 

{a'(ab) ztMI' iP'(ab) Z e(ab)}', 

\a'(ab) z e(ab)}', \y'(ab) z €(aft)}', 

{ T '(g&) Z7M)', 

and upon application of 



Foundations of the Common Logic 9 

(x' z y)' z (/ z *)', 

{•'(oi) Z a(ab)}', W(ab) z a(a&)}', 

[e'(a&) z 0(a&)}', [/(aft) Z 0(a&)}', 

W(ab) Z7MI' {0'(a5) Z aW)' 

We are now prepared to lay down the final generaliza- 
tions which are given below. From the propositions that 
have just been enumerated there will follow 

{w'(ab) z k(ab)}', V 

from III and V, by 

(x z y) z (/ z O, 
(x / y) f z (y f z x'V 
£"(a&) jrw'(ab), ' \w\ab) zk"(ab)}', VI 

from III and IV, by 

z z)'(y z z) z (x z y)\ 
(x / "vV / (y f / x fs ) f 
{k(ab) z w"(ab)}', " \w"(ab) z k{ab)}'. VII 

§ 11. In order to classify the categorical forms under the 
heads, contradictories, contraries, subcontraries, and sub- 
alterns, let us consider what special meanings of x{ab) and 
y(ab) render true or untrue, 

(1) x{ab) Z y'(ab), 

(2) y'iflb) z x(ab). 

If x{ab) and y(ab) satisfy (1) and (2) together, x{ab) is 
said to be contradictory to y{ab). By I, k r (ab) is contra- 
dictory to k(ab) and, by II, k(ab) is contradictory to k'{ab). 

If x(ab) and y(ab) satisfy (1) alone, x{ab) is said to be 
contrary to y(ab). By III and V, k(ab) is contrary to w(ab) . 

If x{ab) and y(ab) satisfy (2) alone x{ab) is said to be sub- 
contrary to y{ab). By VI, k'{ab) is subcontrary to w'(ab). 

If x(a&) and y{ab) satisfy neither (1) nor (2), x(ab) is 
said to be subaltern to y(o5). By VII, k(ab) is subaltern 
to w'(ab), and, by IV, &(aft) and h'{ab) are each the sub- 
alterns of themselves. 



io Non-Aristotelian Logic 

§ 12. In order to classify terms under the heads, contra- 
dictories, contraries, subcontraries and subalterns, let us con- 
sider what special meanings of a and b render true or untrue, 

(1) a'(ab') Z y(ab'), 

(2) a'(b'a) Z y(Va). 

The postulate a {a 'a') z a(a'a') implies a(a'a') z y(a f a'), 
for 

{a'(aV) z a(c'a')| Z (a'(cV) z«'W)l, 

by Z 30 Z (/ Z * ') ; 

{a'(a'a') z«'W)| Z (a'(flV) ZtW), 

by (x z #') Z (# Z y). 

If contradictory terms be those meanings of a and 6 that 
render (i) and (2) true together, then, by a' (a' a') z y(a'a'), 
it follows that a and a' are contradictory. 

If contrary terms be those meanings of a and 6 that cause 
(1) alone to become true, then if we assume, 

a'(oi) z y(oi), 
{a'(io) z y(io)Y, 

where o' = i, i' — 0, we derive in particular the fact that 
o is the contrary of itself. 

If subcontrary terms be those meanings of a and b that 
cause (2) alone to become true, then, from the assumptions 
just written down, it follows in particular that i is the 
subcontrary of itself. 

If (1) and (2) remain untrue for some special meaning 
of a and b, then a is said to be the subaltern of b. From the 
two equivalent propositions, 

{a\aa f ) Z y{aa'))', {a{a f a) z y(a'a)}', 

whose untruth may be established on making a = i, a' = 0, 
in the first, a' = i, a = o, in the second, and which imply 

{a'(ab') ZtW)', W{b'a) z y(b'a)}\ 

it will be seen that a is in general the subaltern of b and 
of itself. 



Foundations of the Common Logic ii 

§ 13. If the meaning of zero (0) is unique; that is, if 
we assume, 

i z {a'(io) / tWT, 

which is the same as, 

{a(io) Z 7 (to)} Z o, 
we should have, 

ot(io) Z 0, y(io) Z o, 
and from 

a!(oi) z y(oi), 
a(oi) z o, y f (oi) z 0. 

These last results and others, that have gone before 
(viz., Ill), yield: 

a' (00) z 0, a (oi) z 0, a(io) z o } a'(ii) Z 0, 

(3 (00) z 0, (oi) z 0, j8(*'o) Z o, (ii) z 0, 

y (00) z 0, t'00 Z 0, TO) Z <?> 7 (*0 Z 0, 

e (00) Z 0, € (0*) z 0, e(«?) Z 0, e (ii) Z 0. 

We may now establish 

{a'(aa') /. a(aa')}', {e'(aa') z e(aa')}', 

{y'(aa') z 7(0*')}', \y(aa') z 7'(aa')}'> 

and if we postulate 
(xiii) a(aa') z ci'iaa 1 ), 

(xiv) 0(oa') Z ffifld), 

(xv) {eOa') Z€ , (aa , )} / , 

(xvi) { 7 '(aa') Z 7(^)}', 

the truth or untruth of every remaining variety of imme- 
diate inference, x(a, b) z y(a, b), may be derived. 

§ 14. The operation of simple conversion consists in the 
interchange of subject and predicate. From y(io) z and 
y'(oi) z 0, which are imposed upon us by the definition 
of the null-class (0)*, it will appear that the inconvertibility 
of y is fundamental ; for 

{i Z o} ; {y(io) z 0} z \i Z y(io)\' 

* The null-class (0) is to be understood as the class that contains no objects, 
or none of the objects that are in question. The universe (i) is the class that 
contains all of the objects that are in question. 



12 



Non-Aristotelian Logic 



by (x z z)'(y z s) z (x z y)\ and 

[i z 7(0*)} {* z 7(^)1' z (tW z y(io)}' 

by (.v Z ?)(* Z *)' z (y Z *)'. 

.*. 7(a&) z 7(^0) is untrue. 



Exercises 
i . The meaning of logical equality is given by 

(x z y)(y Z x) z (* = y), 
(* = y) z Z y)(y Z x). 

If &(a&) = k(ab)k(ab) and &(a&) Z w'(ab), show that 

a(oft) Z P'(ab)y'(ab)e'(ab)y'(ba), 

P(ab) Z a^dbWiafyJiabWQa), 

y(flb) Z a'(ab)p'(ab)e'(abW(ba) t 

e(ab) Z a'(ab)p'(ab)y'(ab)y'(]ba) t 



by the aid of 



2. If 



(* Z dOC? Z ■*) Z (x Z s), 

(* z y) z 0# z sy). 

P'(ab)y'(ab)e'(ab)y'(ba) Z a(o&), 
a'(pb)y'(pb) e'(ab)y'(ba) Z 0(a&), 
a'(ab)p'(ab)e'(ab)y'(ba) Z 7(*&), 
a'(ab)P\ab)y'{db)y'(ba) Z c(o&), 



establish 



a>&) = 0(a&) + 7(06) + e(o6) + y(ba), 

0'(db) = a(a&) + y(ab) + e(a&) + y(ba), 

y'iab) = a(a&) + p(db) + €<a6) + tM, 

e'(ab) = a(a6) + 0(a&) + y(ab) + 7(60), 

assuming that the contradictory of a product is the sum of the 

contradictories of the separate factors and assuming the right 

to substitute k(ab) directly for k"(ab). 

3. Assuming x(ab) = x(ab)x(ab), x(ab) = x(ab) + x(ab), show 
that 

a'(flb)p(flb) = y{ab) + t{ab) + 7(M> etc., etc., 
a(ab) + j8(o6) = y'{aby(ab)y'(ba), etc., etc. 



Foundations of the Common Logic 13 

4. Establish the general results, 

k(ab) - k(ab)w'(ab), k'(ab) = k'(ab) + w(ab), 
kiab)w(ab) = 0, 

5. From the principle, (x z z)'(y z z) Z (x Z y)', and the 
postulate, {a{aa) Z c/(aa)}', derive 

{a(aa) Z p(aa)}', {a(aa) Z y(aa)}', {a(aa) Z e(aa)\'. 

6. By the aid of the principles, 

(x z y){y z z) z (x z z), (x z x') z (x z y), 

from the postulate, a'{aa) Z a(aa), and results already estab- 
lished, (viz., Ill), show that all propositions of the form, 
x(aa) Z y{ao), except the three cases in the last example, are 
true implications, x(aa) and y(aa) representing only the un- 
primed letters. 

7. Show by the method of the last example that a(aa) Z a'(aa) 
is the only untrue implication of the form x(aa) Z y'{aa). 

8. Derive seven true implications of the form, x'(aa) z y(aa)i 
and nine untrue implications of the same form. 

9. Establish the untruth of 

k(a, b) Z w(a, b), k'(a t b) Z w'(a, b). 

10. Establish the untruth of (3(ab) Z P'(ab) and e(ab) Z e'(ab) 
by making them depend upon the untruth of p(ba)fi(cb) z &'(ca) 
and e(ba)e(cb) Z e'(ca) respectively (see the postulates of the 
next chapter). Thus, 

{(3(ba)0(cb) Z Pica))' Z {P{ba)p(cb)p(ca) Z 0}' ', 
by (* Z y'Y Z (xy Z o)'\ 
{p(ba)p(cb)(3(ca) z o} f {0(cb)p(ca)'O z 0} 

Z {Kba)(3(cb)(3(ca) Z /3(cb)l3(ca)-oy, 
by (x;y Z z)'(w Z z) z (xy Z w)'; 

{p(ba)p(cb)p(ca) z P(cb)p(ca)-o}' Z {P(ba) Z 0}' 
by (xz Z zyY Z (x Z y)'; 

Mba) ZoY z Iftba) Z P'(ba)}', 
by (x z 0)' z (x Z x'Y. 



CHAPTER II 

§ 15. A syllogism is an implication belonging to one of 
the types, 

1. x(ba)y{cb) z z(ca), 

2. x{ab)y{cb) z z(ca), 

3. x{ba)y(bc) z z(ca), 

4. x{ab)y(bc) z z(ca). 

These differences are known as the first, second, third 
and fourth figures of the syllogism respectively. The two 
forms conjoined to the left of the implication sign are called 
the premises and the form to the right of the implication 
sign is called the conclusion. The predicate of the con- 
clusion is called the major term and points out the major 
premise, which by convention is written first, while the 
subject of the conclusion is called the minor term and 
points out the minor premise. The term, which is common 
to the premises and which does not appear in the conclusion, 
is called the middle term. 

Since x, y and z may assume any one of the four values, 
a, /?, 7, e, there will be sixty-four ways in each figure, called 
the moods of the syllogism, in which xy /_ z may be ex- 
pressed. True syllogistic variants are called valid moods. 
Those that are untrue are called invalid moods. 

It will be convenient to deduce in the first place all of 
the valid moods of syllogistic form that exist and to estab- 
lish later on the invalidity of those moods that remain. 
In what follows we shall suppose that x, y and z stand 
only for the unprimed letters. Thus: we shall refer to 
x(a, b)y(b, c) z z(ca), x'{a, b)y{b, c) z z'(ca), 

x{a, b)y(b, c) z z r {ca), etc., 

as specific types. 

14 






Moods of the Syllogism 15 

§ 16. The valid moods of the syllogism, 

x(a, b)y(b, c) z z(ca), 

twenty-nine in all, which are not set down among the 
assumptions of § 5 and § 8, may be derived at once by the 
following principles : 

(xy / z)(w z x) z (wy Z z), 
(x z y)(y Z z) z (x z z), 
(xy z z) z (yx Z z). 

Thus, from postulate (x), by the second principle, 
{e(ba)y(cb) z e(ca)}{e(ca) z e(ac) ) z {e(ba)y(cb) Z e(ac)}, 
and, since the term-order in the conclusion is now reversed, 
so that the major term has become the minor term and 
the minor term has become the major term, it will be 
necessary to employ the third principle to restore the 
normal order of the premises. Accordingly, 

{e(ba)y(cb) Z e(ac) } Z \y(cb)e(ba) z e(ac)}, 

and it will be seen that the term order in this result is that 
of the fourth figure. The second principle (above) thus 
enables us to convert simply in the conclusion and the effect 
of simple conversion in the conclusion is to change the first 
figure to the fourth. 

Similarly, since the third principle enables us to arrange 
the premises in either order, the first principle will allow us 
to convert simply in either premise, if that premise be not 
in the 7-form. Thus, from postulate (i) of § 5, 

{a(ba)a(bc) z cx(ca)} z {a(bc)a(ba) z a(ca)} t 

by the third principle (above) ; 

{<x(bc)a(ba) z a(ca) } {a(cb) z a(bc)} 

Z {a(cb)a(ba) z a(ca)}„ 
by the first principle (above) ; 

{a(cb)a(ba) z a(ca)} z \a{ba)a(cb) z ot(ca)}, 

by the third principle (above) ; and this result is a valid 
mood of the first figure. However, when it is desired 



1 6 Non- Aristotelian Logic 

to convert simply in the minor premise, it will be more con- 
venient to employ at once the principle, 

(xy z z)(w / y) z (xw z z), 

and avoid two of the three steps, that would otherwise be 
necessary. 

Exercise 

From postulates (i-iv) of § 5 and postulate (x) of § 8 derive 
twenty-three valid moods of the syllogism by the aid of the 
principles, 

(xy Z z)(w Z x) Z (wy Z 2), 

(xy Z z)(w z y) Z (xw Z 2), 

fry Z z)(z Z w) z (xy Z w), 

(xy Z z) z (yx Z z). 

§ 17. The valid moods of the syllogism, 

x(a, b)y(b, c) z z'(ca), 

one hundred and forty-two in number, as well as those of the 
syllogisms, x(a, b)y f (b, c) z z'(ca) and x'(a, b)y(b, c) z z'(ca), 
which number thirty-one and twenty-seven respectively, 
may now be obtained from the results of § 16 and the forms 
of immediate inference contained in § 8, by the aid of the 
additional principles, 

(xy z z') z (xz z y'), 
(xy z z') z (zy z x'). 

The examples, which follow, will be enough to illustrate 
the method. 

(1) ItMtW Z tW) Z {y(ba)y'(ca) z ?'(«*)}, 
by (xy Z z) z (xz r z /)• 

(2) {y(ab)y'(cb) z y'(ca)}Hcb) z y\cb)) 

Z \y(ab)e(cb) z y'(ca)}, 
by (xy Z z)(w z y) Z (xw z z). 

(3) \y(ab)e(cb) z y'(ca)} z {y(ab)y(ca) z e'(cb)) 
by (xy Z z') z (xz z y'). 



Moods of the Syllogism 17 

No other valid moods of syllogistic form exist, except 
those that have now been enumerated, as will appear in 
the sequel, when all of the remaining variants shall have 
been declared untrue. 

Exercises 

1. From postulate (x) of § 8 above deduce six valid implica- 
tions of the form, x(a, b)y'(b, c) z z'(ca). 

2. From postulate (x) of § 8 above deduce thirty- three valid 
implications of the form, x(a, b)y(b, c) Z z'(ca). 

3. Assuming the special conditions mentioned at the end of 
§ 8 to hold true, show that postulate (ix) of § 8 yields only thirteen 
valid implications of the form given in the last exercise. 

§ 18. It will be convenient in establishing the invalid 
moods of the syllogism to begin with the form 

x(a, b)y(b f c) z z'(ca). 

Any invalid mood under this head, which contains an 
ce-premise or an ce-conclusion, may be shown to be invalid 
by identifying terms in the a-form. Thus: 

1. Suppose a(ba)y(cb) z y'(ca) were valid, and identify 
terms in the major premise. 

{a(aa)y(ca) z y'(ca)}{i z a(aa) } Z {y(ca) Z y'(ca)}, 

by (xy z z)(w z oc) z (wy z z). 

.'. a(ba)y(cb) z y'ica) is invalid. 

2. Suppose y(ab)y(cb) z a'(ca) were valid and identify 
terms in the conclusion. 

{y(ab)y(ab) z a{aa)}{a'(aa) z 0} z (y(ab)y(ab) z 0}, 

by (xy z z)(z z w) z {xy Z w); 

{y(ab)y(ab) z 0} z {y(ab) z y'iab)}, 

by (xy z 0) z (x z y'). 

.'. y(ab)y(cb) z a f (ca) is invalid. 



IS 



Non- Aristotelian Logic 



Exercise 

Establish the invalidity of the thirty-four moods of the syl- 
logism, x(a, b)y(b, c) z z'(ca), which are invalid and which 
contain an a-premise or an a-conclusion. 

In order to deduce the invalid moods, which remain, 
eighty in all, it will be necessary to add eleven postulates 
to the ones already set down. These assumptions are: 



(xvii) 


P(ba)p(cb) / P(ca)}\ 


(xviii) 


{ P(ba)p(cb) z y'(ca)}', 


(xix) j 


P(ba)p(cb) z e'(ca)}', 


(xx) 


y(ab)y(cb) z P'(ca)}', 


(xxi) 


y(ba)y(bc) z (3'(ca)}', 


(xxii) 


y(ba)y(cb) z y'(ca)}', 


(xxiii) 


y(ab)y(cb) z e'(ca)}', 


(xxiv) 


< e(ba)y(bc) z P'(ca)}\ 


(xxv) 


e(ba)y(bc) z e'(ca)}', 


(xxvi) 


e(ba)e(cb) z P{ca)}\ 


(xxvii) 


e(ba) e(cb) z e'(ca)}'. 



Exercise 

From postulates (xvii-xxvii) deduce sixty-nine other non- 
implications of the same form, by the aid of the additional 
principles, 

(xy Z z)'(w Z z) z (xy z w)', 

(xy z z)'(x z w) Z (wy Z z)' , 

(xy Z z)'(y z w) Z (ocw Z z)', 

(xy Z z')' z (xz z y)', 

(xy Z z'Y z (zy Z *')', 

(xy Z z)' z (yx Z s)'. 

§ 19. All of the invalid moods of the syllogistic form, 
x(a, b)y(b, c) z z(ca), x'(a, b)y(b, c) z z'(ca) and 

x(a, b)y'(b, c) z z'(ca)> 

may be deduced at once from the results that have now 
been established. A few examples will be enough to 
illustrate the method. 



Moods of the Syllogism 19 

[e{ba)y{bc) z (3'{ca)}' z [P(ca)e(ba) Z y'{bc)}\ 
by {xy z O' Z {zx Z /)'; 

IfiiabWcb) Zy'{ca)}'{0{ob) zy'{ab)} 

Z {7'(a«€(^) ZT'Wi', 
by (x;y z s)'(# Z w) Z (wy z z)'; 

W{ab)e{cb) Zy'{ca)'\ z {e{cb)y{ca) ZtM)/ 
by (x'y z z')' z (yz z x)'\ 

h'{ab)e{cb) z y f {ca)}' z ly{ca)y'{ab) z e'{cb)}\ 
by {xy Z z')' z {zx z y'Y; 

\e{ba)y{bc) z p f {ca)Y{e{ca) z $'{ca)} 

Z {e{ba)y{bc) z e{ca)}', 

by {xy z z)'{w z z) z {xy z w) f \ 

{e{ba)y{bc) z e{ca)}' ' {e{ac) Ze{ca)} z [y{bc)e{ba) Ze{ac)}', 

by {xy z z)'{w Z z) z {yx z iv)'- 

Exercises, 

1. Show that there exist no valid implications of the form, 
x'{a, b)y{b, c) z z{ca) or x{a, b)y'{b, c) Z z(ca), and consequently 
none of the form, x'{a, b)y f {b } c) z z{ca) or x'{a, b)y'{b, c) Z z'{ca). 

2. Show that as a result of a complete induction of the moods 
in question, {a) a valid mood of the syllogism, whose premises 
and conclusion are all unprimed forms and one of whose premises 
is of the same form as the conclusion, will remain valid, when 
the other premise is put in the a-form ; and (b) a valid mood of 
the syllogism, whose premises are unprimed forms and whose 
conclusion is a primed form and one of whose premises is of a 
different form from the conclusion, will remain valid, when the 
other premise is put in the a-form. 

§ 20. It will be well at this point to indicate the equiva- 
lence of the logic, whose system has now been partially 
developed, with the classical science, perfected in the 
Organon of Aristotle. 

The four categorical forms employed by the traditional 
logic and denoted by the letters, A, E, I, 0, are: 



20 Non-Aristotelian Logic 

A(ab) = All a is b, 
E(ab) = No a is b, 
I(ab) = Some a is b, 
0(ab) = Some a is not b, 

the word some, which is expressed before the subject of I 
and and understood before the predicate of A and I, 
being interpreted to mean, some at least, possibly all. 

This set of four forms satisfies certain conditions, which 
are characteristic of the system, i.e., 

i. Corresponding to each member of the set, there is 
another, which stands for its contradictory; 

2. The relation of subalternation, A(ab) implies I{ab), 
holds true; 

3. A{ab) becomes true, when subject and predicate have 
been identified ; 

4. The subject and predicate of E(ab) and I(ab) alone are 
simply convertible, 

We should have to have, 
to satisfy (1), A(ab)0(ab) z 0, 

E{ab)I(ab) z 0, 

A'{ab)0'{ab) z 0, 

E'(ab)I'(ab) z 0; 
to satisfy (2), A{ab)E{ab) z 0; 

to satisfy (3), A'{aa) z 0, 

to satisfy (4), I(ab) z I(ba)* 

Today it is all but universally taken for granted that 
not all of these conditions hold true, if the terms are allowed 
to take on the meanings nothing and universe, and it is 
usual to retain (1), (3) and (4) and to assert that the relation 
of subalternation of the classical logic is false. This modern 

* It would in the end economize assumptions to take E(ab)A(cb) Z E{ca) 
for granted instead of this last form of immediate inference, for 

E(ab)A{cb) z E(ca) yields E(ab)A{bb) / E(ba), for b = c; 
lE(ab)A(bb) z E{ba)){i z A(bb)} z {E(ab) z E(ba)}, by (3); 
{E(ab) z E(ba)} z {E'(ba) Z E'(ab)}; 

{I(ba) z E'(ba)\{E'(ba) z E\ab)) z [I{ba) Z E'(ab)}, by (1); 
\I(ba) z E'{ab)}{E\ab) z I(ab)} Z [I(ba) Z I(ab)}, by (1). 



Moods of the Syllogism 21 

tradition is, however, based upon a misapprehension, upon 
the supposed necessity of retaining A(ab) = E(ab'). The 
implications set down in the following table will be seen 
to satisfy not only conditions (1-4) but also the definition 
of the null-class and the consequences, that follow from 
that definition. These implications are: 

A' (00) z 0, A f (oi) / 0, A(io) / 0, A'(ii) z 0, 

E(oo) z o, E(oi) z 0, E(io) z 0, E(ii) z 0, 

I (00) z 0, I(oi) z 0, I'(io) Z o, I'iii) z 0, 

0(oo) z o, 0{oi) z o f 0'(io) z 0, 0(ii) z 0. 

We may now state the connection between the traditional 
propositions, A, E, I, O, and the special categorical forms, 
which have been employed in the text. This connection 
is expressed by the following equalities : 

A(ab) = a(ab) + y(ab) f 

E(ab) = e(ab), 

I(ab) = a\ab) + (3(ab) + y(ab) + y(ba), 

0(ab) = e(ab) + (3(ab) + y(ba). 

and it will be readily seen that these values of A, E, I 
and satisfy conditions (.1-4). 

If, now, we express a, (3, y and e in the members of the 
set, A, E, I, 0, we should have 

a{ab) = A(ab)A(ba), 
(3(ab) = l(ab)0(ab)0(ba), 
y(ab) = A(ab)0(ba) t 
e(ab) — E(ab), 

which may be verified by actually multiplying out these 
products as in ordinary algebra and allowing the product 
k(ab)w(ab) to drop out whenever it occurs (see ex. 4 at the 
end of § 14). 

It only remains to be pointed out that these equalities 
and those, which precede them, are satisfied by the implica- 
tions in the table given above and by those contained in 
the similar table in § 13. 



CHAPTER III 

§ 21. In the remarks at the end of § 8 and elsewhere we 
have referred to a system of inference, in which not all of 
the implications of the common logic hold true. It is 
proposed now, as a further illustration of method, to con- 
struct in some detail another system, some of whose char- 
acteristic postulates stand in contradiction to those of 
§ 5 and § 8. Without doing violence to the fundamental 
conditions described in § 12 and § 13, we may assume: 

(i) a'(aa) z a(aa), 

(ii) j8'(aa) z p(aa), 

(iii) y'(aa) Z y(aa), 

(iv) e (ad) z t'(aa), 

which yield at once the equivalent set: 

i z. a(aa), 
i z P(aa), 
i Z y(aa), 
e (aa) z 0. 

%22. In order to frame an image of the possibility of 
a(aa), (3(aa) and y(aa) standing for true propositions, 
imagine the subject-class and the predicate-class, a and b, 
to approach connotative identity. It will then be under- 
stood, how it might become a question, as to whether 
a(ab) f P(ab) and y(ab) are to be regarded as true or false 
in the limiting case. This image would, of course, not 
serve to guide us, unless the assumptions we have made 
had an analytic justification. It is, too, in a sense mis- 
leading, for we shall have to conceive of fi{ab) and y(ab) 
becoming empirically untrue for special concrete meanings 
of a and b that render a(ab) empirically true, without 
making it impossible to regard (3(aa) and y(aa) as true 

22 



Construction of a Non-Aristotelian Logic 23 

for all meanings of a. In interpreting (3(aa) and y(aa), 
the part of a, which is not a, is taken to be the null-part. 
§ 23. The other postulates, by the aid of which we shall 
effect our solution, are: 

(v) p(ab) Z P(ba), (xi) {a(ab) z P(ab)\', 

(vi) P(ab) Z e'(ab), (xii) {a(ab) z y(ab) }', 

(vii) a(ba)a(bc) Za(ca), (xiii) {7(06) Z P(ab)}', 

(viii) a(ba)e(bc) z e(ca), (xiv) {P(ba)(3(cb) Ze'(ca)\', 

(ix) tMt(^) Z7W» (xv) (T^T^Ze'MI', 

(x) y(ab)e(bc) z efca), (xvi) {(3(ba)y(cb) /_e'(ca)}' f 

(xvii) {€(fta)e(c6) Z e'(ca)}'. 

Exercises 

1. Employing the method of § 5, deduce six valid moods of the 
form x(a, b) z y(a, b). 

2. Deduce twenty- two invalid moods of the form given in 
the last example. 

Thus, suppose y(ab) Z y(ba) were valid, 

ItW Z y(ba)\ Z {7(^)7^) Z y(ba)y(cb)}, 
by (x z y) Z (w# z ^:y); 
{7(^)7^) z tMtWHtMtW z €'(**)} 

Z \y{ab)y(cb) Z e'M}, 

by (x Z y){y Z z) Z (x Z z) and a valid syllogism to be ob- 
tained later; but this result is invalid, by (xv); 

.'. y(ab) Z y(ba) is invalid. 

Again, suppose e(ab) Z fiiflb) were valid. 

\e(ba) Z Kba)} Z {e(ba)a(cb) Z /3(&a)a(^)}, 

by (x z y) z (wx z ivy) ; 

{e(ba)a(cb) Z p{ba)a(cb)}{p(ba)a(cb) Z e'(ca)} 

Z {e(ba)a(cb) Z e'(ca)}, 
by the last result, a valid syllogism and 

(x z y)(y z z) z (x z z). 

Now, e(ba)a(cb) Z e(ca) is valid and 

.'. e(ba)a(cb) Z 0, 



:?4 Non-Aristotelian Logic 

by (.v Z y)(x Z /) Z (.v Z o)\ and 

e(ba)a(cb)^(c ] d) Z o, 
by (.v z 6) Z (.v v z o) ; 

.*. e(ba)a(cb)P(cd) Z j8'W» 
by (.v Zo)Z (* Z y) ; 

Identifying c and b and suppressing the a-premise we should 
have 

e(ba)p{bd) Z j8'((to), 
which yields 

P(da)P(bd) Z e'(6a); 

but this result contradicts (xiv) ; 

.'. e(ab) Z j8(a&) is invalid. 

3. Employing the method of § 8 deduce eleven valid moods in 
addition to (vi) of the form, x(a, b) Z y'(fl>, b). 

4. Deduce twenty invalid moods of the form given in the 
last example. 

The only invalid mood of this type, which offers any difficulty, 
is e(ab) z e'(a&). Suppose, then, that this mood were valid. 

U(ba) Z e'(ba)} Z {e(ba)a(cb) Z €'(6a)a(c6)}, 

by (x z y) Z (ww Z wy) ; 

{e(ba)a(cb) Z e'(^)a(^)}{e'(&o)a(c6) Z e'(ca)} 

Z {€(fta)a(c6) Z e'(ca)}, 

by (x Z y)(y Z z) Z (x Z z) and a valid syllogism to be estab- 
lished later; 

{e(ba)a(cb) Z t'(ca)}{e(ba)a(cb) Z e(ca)} Z {e(ba)a(cb) Z 0}, 

by (x Z y)(x Z y f ) Z (x Z 0) and a valid syllogism to be estab- 
lished later; 

{e(ba)a(cb) Z 0} Z {e(ba)a(cb)(3(cd) Z 0}, 

by (x z 0) z (xy Z 0). 

If, now, we identify b and c and suppress the a-premise, we 
should have e(ba)p(bd) Z o\ 

{e(ba)(3(bd) Zo\ Z {e(ba)p(bd) Z P'(da)}, 

by (x Z o) Z (x Z y) ; 

{e(ba)(3(bd) z P'(da)\ Z \p{da)p{U) Z e'(fo)} 



Construction of a Non-Aristotelian Logic 25 

by (xy / z') z (zy Z x') ; but this result contradicts (xiv) above, 
and 

.'. e(ab) Z e'(ab) is invalid. 

5. Establish the invalidity of all the thirty- two moods of the 
form, x'(a, b) Z y(a, b). 

If we assume 0(oi) z to be true, the results of the following 
table will then be forced upon us by what has gone before. 
That is, 

0/(00) Z 0, ol (oi) Z 0, a(io) Z 0, a(ii) Z 0, 

/3'(00) Z 0, j8 (0/) Z 0, /3(**0) Z 0, jS'W Z 0, 

7'(<w) Z <?> y'(oi) Z 0, t(^) Z 0, 7'W Z 0> 

e (00) Z 0, e (fli) Z 0, e(io) Z 0, € (w) Z 0, 

and it will be seen that postulate (xiii) above may now be saved. 

6. Derive thirteen valid moods of the syllogism, 

x(a, b)y(b, c) z z(ca), 

as in § 16, from postulates (viii-x) above. 

7. Prove that two hundred and eleven of the invalid moods 
of the type given in the last exercise may be shown to be invalid 
by the aid of the characteristic postulates (i-iv) above. 

8. Show that the twenty-eight invalid moods not accounted 
for in the last exercise may be made to depend on one or the 
other of postulates (xiv-xvii) above. 

9. Derive the eighty-one valid moods of the syllogism, x(a, b) 
y(b, c) z z'(ca). 

10. Prove that one hundred and forty- four of the invalid moods 
of the type given in the last exercise may be reduced to invalid 
forms of immediate inference already established, and so shown 
to be invalid, by the aid of the characteristic postulates (i-iv) 
above. 

11. Show that twenty-seven invalid moods not accounted 
for in the last exercise may be made to depend on postulates 
(xiv-xvii) above. 

12. As in exercise (1), § 19, show that no other valid syllogistic 
variations exist, except those contained in the syllogisms, 
x'(a, b)y(b, c) /_ z'(ca) and x(a, b)y'(b, c) z z'(ca). 



CHAPTER IV 

§ 24. A sorites is a form of implication of the general 
type,* 

x(i, 2)y(2, 3) ••- z(n — J, n) z w(ni). 

Certain valid moods of the sorites can be constructed from 
chains of valid syllogisms. For example the chain, 

y(2i)y(32) z 7(31), 

y(3ih(43) Z 7(41), 

y(4i)y(54) z 7(51), 
will yield a valid mood, viz., 

7(21)7(32)7(43)7(54) Z 7(5 J)» 
for 

{7(41)7(54) z. 7(51)} {7(31)7(43) /#)) 

Z {7(31)7(43)7(54) Z7(5J)}» 
and 

{7(31)7(43)7(54) z. 7(51)} {7(21)7(32) zt(jj)} 

Z {7(21)7(32)7(43)7(54) Z7(5')}. 
Again, 

{a(2j)a(j2) Z«(JJ)} Z {a(27)a(j2)a(4j) Za(j/)a(^j)}. 
.'. a(2l)a(32)a(43) Z a(^r). 

The valid mood of the sorites is, accordingly, built up 
out of the chain, 

a(2l)a(32) Z a(3l), 

a(3l)a(43) Z a(4l). 

It remains to be proven that the only valid moods of the 
sorites that exist can be built up from chains of valid syllogisms. 

It will be convenient to take the conclusion successively 
in each one of the eight possible forms. 

* In what follows it will be convenient to employ the ordinal numbers for 
class-terms instead of the initial letters of the alphabet. The solution given 
here belongs to the logic of the last chapter. 

26 



The General Solution of the Sorites 27 

Conclusion a' and All Premises Unprimed 

Suppose in the first instance that no e-premise is present. 
Then no a-premise is present, for suppose x(s, s + 1) to 
be an a-premise. Identifying terms in the a-, /?- and 
7-premises except x, the mood of the sorites will reduce 
to an invalid mood of immediate inference, viz., 

a(s, s + i) Z a'(s, s + l). 
Similarly no /3-premise can occur, by 

P(s, s + 1) z ot'(s, s + 1), 
and no 7-premise can occur, by 

7(5, S + i) Z a'(s, S + I). 

Consequently at least one €-premise is present if the mood 
of the sorites is valid. 

Not more than a single e-premise can be present, for 
if two or more e-premises were present, the mood of the 
sorites could, by identifying terms and suppressing the a-, 
/?- and 7-premises, be reduced to the form, 

e(l, 2)e(2, 3) • • • e(n - J, n) Z a' (til), 

and the validity of this mood can be made to depend on 
that of a mood in which all but two of the premises are 
absent,* viz., 

e(>, S + l)e(s + J, $ + 2) Z ol{s + 2 s), 

which is invalid. 

Contradict the €-premise and the conclusion and inter- 
change and the sorites reduces to the case of an e'-conclu- 
sion, which will be considered later on. 

* Thus, e(l2)e(23)e(34)e(4S) L a'isi) yields e(i4)e(43)e(34)e(4S) Z <x f (jl) 
for 2=4, and 

{e(l4)e(43H34)<45) Z a'^i)} {^34) Z e(43H34)} 

, . . . Z {e(i4h(34h(45) Z«'(5i)}. 

Thus we obtain in succession, 

e(i4h<34)*(45) ^ oc'(5i), for 2 = 4, 
<34)*(45) Z «'(5j), for 1 = 3. 



28 Non-Aristotelian Logic 

Conclusion j3 f or y' and All Premises Unprimed 
Exactly as in the last case it can be shown that at least 
one e-premise must be present and that not more than one 
e-premise can occur. Contradicting and interchanging as 
before the e-premise and the conclusion, the sorites reduces 
to the case of an e'-conclusion. 

Conclusion e' and All Premises Unprimed 
No e-premise can occur, for, if one or more e-premises 
were present, the mood of the sorites would reduce to the 
form, 

e(z, 2)e(>, 3) • • • e(n - I, n) z e(ni), 

by identifying terms in all of the a-, /?- and y-premises, 
and this form is reducible to the invalid syllogism, 

e(>, 5 + l)e(s + I, S + 2) Z e'O + 2 s), 

or an invalid mood of immediate inference, 

e(s, s + i) Z e'(s S + j). 

Not more than a single /^-premise can occur, for, if two 
or more /3-premises were present, the mood of the sorites 
would, by identifying terms in all of the premises but two 
of the /^-premises, be reducible to an invalid syllogism of 
the form, 

(3(s, s -f i)p(s + I, s + 2) z e'(s + 2 s). 

Suppose that no /?- or y-premise is present. Then all of 
the premises are in the a-form and the sorites becomes, 

a(l, 2)a(2, 3) • • • a{n - J, n) Z e'(ni), 

which can be constructed from the chain of valid syllogisms, 

a(l, 2)a(2, 3) Z a(3l), 
a(3l)a(3, 4) Z a(4l), 



a(n — 2 i)a(n — 2, n — 1) z a(n — I 1), 
a(n — i i)a(n — i, n) z t'(n 1). 

If no y-premise is present and all but one /3-premise is 
in the a-form, the sorites then becomes, 



The General Solution of the Sorites 29 

a(i, 2)a(2, 3) • • • a(s — /, s)|80> 5 + l)a(j + J, 5 + 2) 

• • • a(n — 1, n) / e'(wi), 

which can be built up from the chain of valid syllogisms, 

a(l, 2)a{2,3) / a(jl), 
a(3l)a(3,4) / a(4l), 

a(s — I l)a(s — I, s) Z a(si), 
a(si)(3(s, s + l) Z e'O + I j), 
e'O + i" /)«0 + J, * + 2) Z e'O + 2 i), 



e'(n — J j)a(w — I, n) Z e'(ni). 

Suppose, again, that all of the premises are in the y-form, 
i.e., that the sorites is 

y(i, 2)7(2, 3) • • • y(n - 7, rc) z e'(rcj). 

The first premise, which presents the term-order (s — 1 s), 
i.e., with the smaller ordinal number appearing as subject, 
establishes that order in each one of the premises which 
follow. For, suppose that the term order (s — 1 s) , having 
once occurred, should appear reversed later on. The 
sorites would, by identifying terms, be reducible to an 
invalid syllogism, viz., 

7O ■*■ I s)y(s + 1 s) z e'O + 1 s - l). 

The sorites becomes, consequently, 

y(2l)y(32) • • • 7O 5 - i)y(s s + 1) 

• • • y(n — in)/. e'(m), 

which can be derived from the chain, 

y(2l)y{32) z t(j/), 

7(31)7(43) z 7(41), 



y(si)y(s s + 1) z e'O + 11), 

e'O + I l)rO + I s + 2) Z e'O + 2 i), 

e'(n — i i)y(n — in)/. e'(m). 
If the mood of the sorites contains only a- and 7-premises, 



30 Non-Aristotelian Logic 

the term order in each 7-premise is established as above, 
the first 7-premise, which presents the term order (s — is)* 
establishing that order in each 7-premise, which follows. 
The generating chain of syllogisms will be the same as 
the last, except that each minor premise in the chain, 
which corresponds to an a-premise of the sorites, will 
appear in the a-form. 

If all of the premises, except a single /3-premise, be in 
the 7-form, the sorites becomes, 

7(7, 2)7(2, 3) • • • y{s - 1, s)(3(s, s + i)y(s + I, s + 2) 

• • • y{n — 1, n) z e'(m). 

The term-order in each premise, which precedes the /?- 
premise, is established as (s s — 1) . For, if the term-order 
in a 7-premise coming before the /3-premise should appear 
as (s — 1 s), then, by identifying terms, the mood of the 
sorites would be reducible to an invalid syllogism of the 
form 

7O - I s)@(s, s + l) Z e'O + I s - I). 

Each premise, which follows the /3-premise, must present 
the term-order (s — is), for otherwise the mood of the 
sorites would be reducible to an invalid syllogism, vz., 

(3(s — 2, s — l)y(s s — l) Z e'(s s — 2). 

The term-order being now unambiguously established, the 
sorites becomes 

7(27)7(52) ... y(s s - i)j3(s, s + i)y(s + 1 s + 2) 

• • • y{n — 1 n) Z e'(ni), 

which may be generated from the chain, 

7(27)7(52) z y(3i), 
y(3i)y(43) z t(^), 



y(s - 1 i)y(s s - 1) Z y(si), 

7(57)0(5, s + 7) z e'(s + 7 7), 

e'(s + I l)y(s + 75 + 2) Z e'(s + 2 7), 

e'(n — 1 i)y(n — 1 n) Z e'(m), 



The General Solution of the Sorites 31 

or, if the initial premise be in the /?-form, from 
0(1, 2)7(23) Z e'(3i), 
*' (3ih(34) Z *(4i), 



e'(n — I i)y(n — 1 n) z e'(ni). 

The remaining moods of valid sorites which contain a-, 
/?- and 7-premises and an e'-conclusion are obtained from 
the last type by replacing one or more 7-premises by a- 
premises in every possible way. Each type so obtained 
can be constructed from one of these last chains of valid 
syllogisms, except that now the minor premise of each 
member of the chain, that corresponds to an a-premise 
of the sorites, will appear in the a-form. 

There exist, consequently, no valid moods of the sorites, 
in which the conclusion is in the e'-form and in which 
none of the premises is a primed form, which cannot be 
constructed from chains of valid syllogisms. All the other 
moods, in which the premises are unprimed and the con- 
clusion is a primed form, are gotten from the valid moods 
already established by the aid of the principle, 

(xy Lz) z. (ocz f z y'). 

§ 25. Conclusion a and All Premises Unprimed 

It will be easy to show that no (3-, 7- or e-premise can 
occur, if the mood of the sorites is valid, and that, conse- 
quently, the general form of implication will be 

a(l, 2)a(2, 3) . . . a(n — I, n) Z a(ni), 

whose chain of generating syllogisms is 

a(l, 2)a(2,3) Z a(3l), 
a(3l)a(3, 4) Z 01(41), 

a(n — 1 i)a(n — I, n) z a(ni). 

Conclusion /5 and All Premises Unprimed 
Under this head it will be found that no a-, 7- or e- 
premise can occur and that consequently all ol the premises 



32 Non-Aristotelian Logic 

are in the 0-form. But such a sorites may be reduced to 
an invalid syllogism, viz., 

P(s - i, s)0(s, s + i) z P(s + I s - i). 

There exist, consequently, no valid moods of this type. 

Conclusion y and All Premises Unprimed 

Here it can be shown at once that no a-, 0- or e-premise 
can occur and that, consequently, all of the premises are in 
the Y-form. Moreover, the term-order in each 7-premise 
is established as (s s — 1), i.e., with the larger ordinal 
number coming first; for, suppose one of the premises 
should appear as y(s — 1 s). The mood of the sorites 
would then be reducible to an invalid syllogism of one of 
the forms, 

y(s - 1 s)y(s, s + 1) z y(s + I s - 1), 
7(5 — 2, s — i)y(s — 1 s) z y(s s — 2). 

The sorites becomes, then, 

y(2i)y(j2) • • • y{n n - 1) z tO *}• 

and its chain of generating syllogisms is 

y(2l)y(32) Z 7(jl), 
7(31)7(43) Z y(4i)> 



y(n — 1 i)y(n n — 1) z y(n 1). 

Conclusion e and All Premises Unprimed 
Just as in the cases already considered, it will be easy 
to show that one and only one e-premise must be present 
and that no ^-premise can occur. One form of this sorites 
is, accordingly, 

a(l, 2)a(2, 3) ••• a(s - J, s)e(s, s + l)a(s + J, s + 2) 

• • • a(n — J, n) z e(m), 

which can be constructed from the chain of valid syllogisms, 



The General Solution of the Sorites 33 

a(l, 2)a(2,j) z a(3l), 
a(3l)a(3, 4) / 01(41), 



a(s — I l)a(s — I, s) Z a(si), 
a(si)e(s, s + i) Z e(s -f- I i), 
e(s + I l)a(s + I, 5 + 2) Z c(s + 2 i), 

e(n — I l)a(n — I, n) Z e(ni). 

The other valid sorites of this type are gotten by re- 
placing one or more a-premises, coming before the e- 
premise, by a y(s — is), one or more a-premises, coming 
after the e-premise, by a y(s s — /), and it will be easy 
in each case to construct the generating chain of syllogisms. 
There exist, consequently, no valid moods of the sorites, 
whose premises and conclusion are all unprimed forms, 
which cannot be built up from chains of valid syllogisms. 
It only remains to be shown that the valid types already 
established are the only valid types that exist, except 
those immediately derived from these by the principle, 
(xy z z) z (xz' £y'). 

All valid moods of the sorites, in which the conclusion 
is a primed form and a single one of the premises is a primed 
form, follow at once from the valid moods already estab- 
lished by the aid of the principle (xy z z) z ,(xz r z y')- 
For, suppose a valid mood of this type, but not so derived, 
should exist. Then a mood of the sorites already estab- 
lished as invalid would appear as valid upon application 
of the same principle, i.e., (xz' z y') Z (xy" z z"), or 
(xy' z z') z (xz z y). 

No valid implications exist, in which a single premise is 
a primed form and the other premises and conclusion are 
unprimed forms. Suppose in the first instance that the 
conclusion is a or y. Then if the primed premise is e it 
may be strengthened* to an unprimed /3-premise and, if 
the primed premise be a, 13 or 7, it may be strengthened 

* If x z y, then x is said to be a strengthened form of y and y is said to be 
a weakened form of x. 



34 Non-Aristotelian Logic 

to e. In either case the resulting mood is one already 
shown to be invalid. If the conclusion is £ and the primed 
premise be strengthened to any unprimed premise, the 
resulting mood is invalid. If, finally, the conclusion is e, 
any primed e-premise may be strengthened to (3, any 
primed a-, (3-, or 7-premise to e, if another e-pemise is 
present, and the resulting mood is again invalid. If the 
conclusion is e and no €-premise is present the mood will 
reduce to x(s — 1, s) y'(s, s + 1) Z e(s+ 1 s — 1). Continu- 
ing this same line of reasoning, it will be seen that no valid 
moods of the sorites exist, in which the conclusion is un- 
primed and two or more premises are primed. 

Finally no valid moods of the sorites exist, in which the 
conclusion is a primed form and in which two or more of 
the premises are primed forms. For suppose such a mood 
to exist. Then, by contradicting and interchanging one 
of the primed premises and the conclusion, the validity 
of a mood already found to be invalid would follow. 

All of the valid implications of the general form, 

x(i, 2)y(2, 3) - - • z{n — 1, n) z w(ni) f 

x • • • z, w f standing for either primed or for unprimed 
letters, have, accordingly, been established, without intro- 
ducing any assumptions except those essential, to the 
solution of the forms of immediate inference and of the 
syllogism. This rather general type of inference may be 
expressed conveniently in the form of the product of n 
premises containing a cycle of n terms and implying zero, 
thus: 

x(i, 2)y(2, 3) • • • z(n, 1) z 0. 

That the solution of this last type is exactly equivalent to 
the solution just given follows from the principles, 

(x z y) Z (xy f z 0), 

(xy' z 0) z (x z y). 

Exercises 

1. Construct a valid mood of the sorites from the chain of 
syllogisms, 



The General Solution of the Sorites 35 

a(2l)y(32) Z y(3*)> 

y(3i)a(43) Z 7(4'), 
f y (4i)yi54) Z y(5i)> 

which are valid in the common logic (§§ 1-19). 

2. If eye (first and second figure) and yee (second and fourth 
figure) be regarded as invalid moods of the syllogism (see the 
concluding remarks of § 8) establish the invalidity of the sorites, 

y(l2)y(23) • • • y(s - I s)e(s, s + i)y(s + 2i + l) 

• • • y(n n — 1) /_ e(ni), 
by the aid of the following 

Principle. — A valid mood of the sorites, whose premises and 
conclusion are all unprimed forms and which has one premise 
of the same form as the conclusion, will remain valid, when as 
many other /3- and y-premises as we desire, are put in the a-form. 

3. Prove, that there exists no valid mood of the sorites, in 
which none of the unprimed premises has the same form as the 
unprimed conclusion, by the aid of the following 

Principle. — A valid mood of the sorites, whose premises and 
conclusion are all unprimed forms and none of whose premises 
has the same form as the conclusion, will remain valid, when as 
many /3- and Y-premises as we desire are put in the a-form. 

4. Employing the principle of exercise 2 reduce the sorites 
a(2i)y(j2)a(4j)y(S4) Z 7.(5-0, which is valid in the common 
logic (§§ 1-19), successively to each one of the three valid syl- 
logisms of exercise 1 . 

5. Employing the same principle, establish the invalidity of 
the sorites, 

y(2i)y(32)y(34)y(54) / 7(51). 

6. From what chain of valid syllogisms (§§ 15-19) can the 
sorites, 

7(12)7(23)6(3, 4)7(54)7(65) Z e(6l), 
be built up? 

7. By the aid of the principle of exercise 2, solve the sorites 
of the common logic for the case, in which all of the premises 
and the conclusion are unprimed forms. 

8. Complete the solution of the sorites begun in exercise 7, 
taking for granted the following principles : 

(a) A valid mood of the sorites, whose premises are all un- 



36 Non-Aristotelian Logic 

primed forms, whose conclusion is a primed form and all of 
whose premises and conclusion are of the same form, will remain 
valid, when as many premises, as we desire, but one, are put 
in the a-form, 

(b) A valid mood of the sorites, whose premises are all un- 
primed forms, whose conclusion is a primed form and one of 
whose premises is a form different from the conclusion, will 
remain valid, when as many other /3- or 7-premises, as we desire, 
are put in the a- form. 



CHAPTER V 

§26. In § 12 are laid down certain conditions, which 
must be taken account of in setting down the foundations 
of any system of inference. The conditions are 

a{io) + y{io) z o, a'(oi)y f (oi) z o, I 

which contains as a consequence a(aa)y'(aa) z o, or in 
particular, 

a (00) y' (00) z 0, a(ii)y'(ii) z 0. II 

Thus, we should have to have 

(a) y{oi) is a true proposition, 

(b) a(oi), a(io) and y(io) are false propositions, 

(c) either a (00) or y{oo) is a true proposition, 

(d) either a(ii) or y(ii) is a true proposition. 

These results, which are forced upon us as a matter of 
definition, leave us a number of choices as to the truth or 
falsity of and e, where subject and predicate are allowed 
to take on the meanings nothing and universe in every 
possible way. 

In order to determine another of these systems, we 
might, by introducing a series of postulates, remove one 
possibility after another, until no choice among alternatives 
remains. As one further illustration of method, we shall 
determine the system, which appears the most paradoxical 
to ordinary intuition, the one, namely, which asserts the 
untruth, for all meanings of a, of the proposition all a is 
all a. 

We shall assume in the first place that a(ab), y{ab) and 
the product, p'(ab)e'(ab), are convertible by contraposition, 
i.e., denoting non-a by a f , 

(1) $'{aby(ab) z Vib'a'ytb'a'), 

a(ab) z a (b'a f ), 
y(ab) z y (b'a f ). 

37 



38 Non-Aristotelian Logic 

Our other postulates will be : 

(2) a(ab) z a'(ab')y'(ab'), 

which yields a(oo) z o, for a = b = o, by I. Consequently, 
y'ipo) z o, by II; a(ii) z o f by (i); y'iii) z 0, by II or 

(i); 

(3) P(ab) Z a'(ab')y'(ab'), 

which yields (3(oo) z o, for a = b = o, by I; and /3(0z) z o, 
P(io) z o, for a = o, b = i, by II; 

(4) «'(*&) Z a'(a&')V(a&'). 

which yields e r (^) z 0, for a = 6 = o, by I ; and t'(oi) z 0, 
c'(^) Z 0, for a = 6 = i, by I. 

(5) a'(ab')y'(ab') z e'(a&), 

which yields e(ii) z 0, for a = b = i, by I. 

The only case, which remains unsettled, is that of p(ii) 
and it may now be seen, from the first member of (i) that 
P'(ii) Z o. For convenience of reference we may now sum- 
marize our results : 

a (oo) Z o a (oi) Z o a(io) z o a (ii) Z o 

P (oo) z o p (oi) z o P(io) Z o p'(ii) z o 

y'(oo) z o y'(oi) z 7(^0) Z o y'(ii) Z 

e' (oo) zo e' (crc) z o e'(io) z o e (ii) z 

It only remains to add to what has gone before, viz., 
a(oo) z o, a(ii) z 0, the more general postulate, 

a(aa) z o. 

Without this postulate it still remains unsettled, whether 
we intend to deny, merely, the truth of all a is all a, or 
to assert its untruth for all meanings of a. 

In the exercises below, it will be taken for granted that 
the forms of immediate inference, which are untrue in the 
common logic, are invalid in the system, whose foundations 
are set down here. 



Alternative Systems 39 

Exercises 

1. Deduce the valid-moods of the syllogism, 

x(a, b)y(b, c) z z(ca), 

which are twenty-one in number, from the following postulates: 

(i) a(ba)P(cb) Z P(ca), (ii) a(ba)e(cb) Z e(ca), 

(iii) y(ba)y(cb) Z y(ca), (iv) y(ab)e(bc) /_ e(ca), 
(v) 0(ab) Z P(ba), (vi) a(ab) Z a(ba). 

2. Deduce the valid moods of the syllogisms, x'(a, b)y(b f c) 
Z z'(ca) and x(a, b)y'(b, c) z z'(ca), of which there are nineteen 
and twenty-three respectively, from the results of exercise 1. 

3. Deduce the valid moods of the syllogism, x(a, b)y(b f c) 
Z z'(ca), there being one hundred and fourteen of this type, from 
the postulates and results of exercise 1, by the aid of the addi- 
tional postulates: 

(vii) a(ba)0(cb) Z y'(ca), 

(viii) P(ba)a(cb) Z y'(ca), 

(ix) a (ba) e(cb) Z y'(ca), 

(x) a(db) Z y'(ab), 

(xi) P{ab) Z e'iab). 

4. Show that the members of the following set may be made 
to depend upon the implications that have already been obtained : 

a(aa) Z a'{aa), \ot!{aa) Z a (aa)}' f 

{p(aa) z ff{a*)Y, {P'(aa) Z (aa)}', 

\y{aa) Z y'(aa)}', y'(aa) z 7 (aa), 

{e(aa) Z e'(aa)}', {e'(aa) Z e(aa)} f , 

IKab) Zt'M!', \y(ab) Z Mob)}', 

{y(ab) Z e'(ab)}', { e(ab) Z y'(ab)}'. 
{y(ab) ZtW)', 

5. Prove that ninety-six of the invalid moods of the syllogism, 
x(a, b)y(b f c) Z z'(ca), may be reduced to simpler invalid forms 
of inference already established and so shown to be invalid, 
(a) either by identifying terms in a 7-premise or a 7-conclusion 
and suppressing the part y(aa), or, (b) by replacing the subject 
and predicate of a |8-premise or a /3-conclusion by unity and 
suppressing the part p(ii). 

6. Show that the remaining forty-six invalid moods not ac- 



40 Non-Aristotelian Logic 

counted for in exercise 5 may be obtained by the aid of the addi- 
tional postulates: 

(xii) {a(ba)a(cb) Z<*'(ca)}', (xiii) \a{ba) e(cb) Z e(ca)}\ 

(xiv) [a(ba)0(cb) Z ff(ca)Y, (xv) {P(ba)(3(cb) Z *'(ca)}', 

(xvi) {a(ba)y(cb) Z t'W)', (xvii) {y(ba)a(cb) Z y'(ca)}'. 

7. Derive the invalidity of all but eight of the two hundred and 
thirty-five invalid moods of the syllogism x(a, b)y(b, c) z z(ca) 
from the results established in exercises 5 and 6. 

8. Employ the results of exercise 7 in order to show that there 
exist no valid syllogistic variations of the form x'(a, b)y{b, c) 
Z z(ca), x(a, b)y'(b, c) Z z(ca), x'(a, b)y f (b, c) z z(ca), or 

x'(a, b)y f (b, c) Z z'ica). 



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